Skip to main content
SpringerLink
Log in

An analysis of the numerical solution of Fredholm integral equations of the first kind

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Summary

This paper analyzes the numerical solution of Fredholm integral equations of the first kindTx=y by means of finite rank and other approximation methods replacingTx=y byT N x=y N ,N=1,2, .... The operatorsT andT N can be viewed as operators from eitherL 2[a, b] toL 2[c,d] or as operators fromL ∞[a, b] toL ∞[c, d]. A complete analysis of the fully discretized problem as compared with the continuous problemTx=y is also given. The filtered least squares minimum norm solutions (LSMN) to the discrete problem and toT N x=y are compared with the LSMN solution ofTx=y. Rates of convergence are included in all cases and are in terms of the mesh spacing of the quadrature for the fully discretized problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anselone, P.M.: Collectively compact operator approximation theory. Prentice-Hall 1971

  2. Anselone, P.M., Palmer, T.W.: Spectral analysis of collectively compact strongly convergent operator sequences. Pacific J. Math.25, 423–431 (1968)

    Google Scholar 

  3. Anselone, P.M., Lee, J.W.: Double approximation methods for the solution of Fredholm integral equations. Proceedings of a Conferrence at Oberwolfach, July, 1975

  4. Anselone, P.M., Davis, J.: Nonlinear operator approximation. Forthcoming

  5. Atkinson, K.E.: The numerical solution of the eigenvalue problem for compact integral operators. Trans. Amer. Math. Soc.129, 458–465 (1967)

    Google Scholar 

  6. Atkinson, K.E.: The numerical solution of Fredholm integral equations of the second kind. SIAM J. Numer. Anal.4, 337–348 (1967)

    Google Scholar 

  7. Baker, C.T.H., Fox, L., Mayers, D.F., Wright, K.: Numerical solution of Fredholm integral equations of the first kind. Comput. J.7, 141–148 (1974)

    Google Scholar 

  8. Brackhage, H.: Über die numerische Behandlung von Integralgleichungen nach Quadraturformelmethode. Numer. Math.3, 147–175 (1969).

    Google Scholar 

  9. Childers, D.S., Varga, R.S., Perry, N.W.: Composite signal decomposition. IEEE Trans. AU-18, No. 4, 471–477 (1970)

    Google Scholar 

  10. Diaz, J.B., Metcalf, E.T.: On iteration procedures for equations of the first kind. Math. Comput.24, 923–935 (1970)

    Google Scholar 

  11. Dunford, H., Schwartz, J.: Linear operators, Part II. New York: Wiley 1963

    Google Scholar 

  12. Gordon, R.: A bibliography on image reconstruction from projections. In: Proc. of Conference on Image Processing at Stanford Univ., 1975

  13. Gordon, R., Herman, G.T.: Three dimensional reconstructions from projections. International Rev. Cytol.38, No. 111 (1974)

    Google Scholar 

  14. Graves, J., Prenter, P.M.: On generalized iterative filters for ill-conditioned problems. To appear

  15. Hanson, R.J.: A numerical method for solving Fredholm integral equations of the first kind using singular values. SIAM J. Numer. Anal.8, 616–622 (1971)

    Google Scholar 

  16. Hanson, R.J.: Integral equations of immunology. Comm. ACM10, 883–890 (1972)

    Google Scholar 

  17. Herman, G.T., Rowland, S.W.: Three methods for reconstructing objects from x-rays: A comparative study. Comput. Graphics and Image Processing2, 151–178 (1973)

    Google Scholar 

  18. Hunt, B.R.: The inverse problem of radiography. Math. Biosci.8, 161–179 (1970)

    Google Scholar 

  19. Kammerer, W.J., Nashed, M.Z.: Iterative methods for best approximate solutions of integral equations of the first and second kinds. MRC Tech. Summary Report 1117, January 1971. J. Math. Anal. Appl.40, 547–573 (1972)

    Google Scholar 

  20. Keller, H.B.: The solution of singular and semidefinite linear systems by iteration. SIAM J. Numer. Anal.2, 281–290 (1965)

    Google Scholar 

  21. Landweber, L.: An iteration formula for Fredholm integral equations of the first kind. Amer. J. Math.73, 615–624 (1951)

    Google Scholar 

  22. MacAdam, D.P.: Digital image restoration by constrained deconvolution. J. Opt. Soc. Amer.60, 1617–1627 (1970)

    Google Scholar 

  23. Mikhlin, S.G.: The numerical performance of variational methods. Wolters-Noordhoff 1971

  24. Nashed, M.Z.: Approximate regularized solutions to improperly posed linear integral and operator equations, pp. 289–322. In: Constructive and computational methods for differential and integral equations (D. Colton, R.G. Gilbert, ed.), Lecture Notes in Mathematics, Vol. 430. Berlin-Heidelberg-New York: Springer 1974

    Google Scholar 

  25. Nashed, M.Z.: On moment discretization and least squares solutions of linear integral equations of the first kind. J. Math. Anal. Appl.53, 359–366 (1976)

    Google Scholar 

  26. Nashed, M.Z.: Aspects of generalized inverses in analysis and regularization. In: Generalized inverses and applications, pp. 193–240, pp. 325–396. New York: Academic Press 1976

    Google Scholar 

  27. Nashed, M.Z., Rall, L.B.: Annotated bibliography on generalized inverses and applications (especially Section L). New York: Academic Press 1976

    Google Scholar 

  28. Nashed, M.Z., Wahba, G.: Generalized inverses in reproducing kernel spaces. An approach to regularization of linear operator equations. SIAM J. Math. Anal.5, 974–987 (1974)

    Google Scholar 

  29. Nystrom, E.J.: Über die praktische Auflösung von linearen Integralgleichungen mit Anwendungen auf Randwertaufgaben der Potentialtheorie. Comment. Phys.-Math. Soc. Sci. Fennica4 (1928)

  30. Phillips, D.L.: A technique for the numerical solution of certain integral equations of the first kind. J. Assoc. Comput. Mach.9, 84–97 (1962)

    Google Scholar 

  31. Picard, E.: Sur un theoreme generale relatif aux equations integrales de premiere espaces et sur quelques problemes de physique mathematique. Rend. Circ. Mat. Palermo29, 615–629 (1910)

    Google Scholar 

  32. Prenter, P.M.: Splines and variational methods. Series pure and applied mathematics, pp. 318. New York: Wiley 1975

    Google Scholar 

  33. Prenter, P.M.: A collocation method for the numerical solution of integral equations. SIAM J. Numer. Anal.10, 571–581 (1973)

    Google Scholar 

  34. Robinson, A.L.: Image reconstruction. I. Computerized x-ray scanners. Science190, No. 542 (1975)

    Google Scholar 

  35. Robinson, A.L.: Image reconstruction. II. Computerized scanner explosion. Science150, No. 647 (1975)

    Google Scholar 

  36. Spence, A.: On the convergence of the Nystrom method for the integral equation eigenvalue problem. Numer. Math.25, 57–66 (1975)

    Google Scholar 

  37. Strand, O.N.: Theory and methods related to the singular-function expansion and Landweber iteration for integral equations of the first kind. SIAM J. Numer. Anal.11, 798–825 (1974)

    Google Scholar 

  38. Strand, O.N.: Theory and methods for operator equations of the first kind. Doctoral thesis, Colorado State University, Fort Collins, 1972

    Google Scholar 

  39. Strand, O.N.: Some aspects of the behavior of regularized solutions as the amount of smoothing is varied. Comput. Math. Appl. (to appear)

  40. Strand, O.N., Westwater, E.R.: Minimum rms estimation of the numerical solution of a Fredholm integral equation of the first kind. SIAM J. Numer. Anal.5, 287–295 (1968)

    Google Scholar 

  41. Tikhonov, A.N.: Solution of incorrectly formulated problems and the regularization method. Soviet Math. Dokl.4, 1035–1038 (1963)

    Google Scholar 

  42. Twomey, S.: On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature. J. Assoc. Comput Mach.10, 79–101 (1963)

    Google Scholar 

  43. Twomey, S.: The application of numerical filtering to the solution of integral equations encountered in indirect sensing measurements. J. Franklin Institute279, 95–109 (1965)

    Google Scholar 

  44. Twomey, S.: The determination of aerosol size distributions from diffusional decay measurements. J. Franklin Inst.275, 121–138 (1963)

    Google Scholar 

  45. Varah, J.M.: On the numerical solution of ill-conditioned linear systems with applications to illposed problems. SIAM J. Numer. Anal.10, 257–267 (1973)

    Google Scholar 

  46. Vermuri, V., Chen, F.P.: An initial value method for solving Fredholm integral equations of the first kind. J. Franklin Inst.297, 187–200 (1974)

    Google Scholar 

  47. Wahba, G.: On the numerical solution of Fredholm integral equations of the first kind. Rep. UWIS-DS-69-217, Univ. of Wisconsin, Madison, 1969

    Google Scholar 

  48. Wahba, G.: Convergence rates of certain approximate solutions to Fredholm integral equations of the first kind. J. Approximation Theory7, 167–185 (1973)

    Google Scholar 

  49. Wahba, G.: A class of approximate solutions to linear operator equations. J. Approximation Theory9, 61–77 (1973)

    Google Scholar 

  50. Wielandt, H.: Error bounds for eigenvalues of symmetric integral equations. Proc. Sympos. Appl. Math. Vol. 6. Providence, RI: Amer. Math. Soc. 1956

    Google Scholar 

  51. Wen so lo: Spectral approximation theory for bounded linear operators. Bull. Austral. Math. Soc.8, 279–387 (1967)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Oregon State University, 97331, Corvallis, OR, USA

    J. W. Lee

  2. Department of Mathematics, Colorado State University, 80523, Fort Collins, CO, USA

    P. M. Prenter

Authors
  1. J. W. Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. M. Prenter
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lee, J.W., Prenter, P.M. An analysis of the numerical solution of Fredholm integral equations of the first kind. Numer. Math. 30, 1–23 (1978). https://doi.org/10.1007/BF01403903

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01403903

Subject Classifications

Search

Navigation